Fig. 1

Schematic circuit implementation of the experimental cycle benchmarking protocol. The protocol can be subdivided into three parts, depicted by the different colors. The green gates \(\tilde{{\mathcal{B}}}\) describe basis changing operations for the state preparation and the measurement (SPAM) procedure. The red gates \(\tilde{{\mathcal{G}}}\) are the noisy implementations of some gate of interest (in this work, the global Mølmer–Sørensen gate acting on all qubits). The blue gates are random Pauli cycles that are introduced to create an effective Pauli channel per application of the gate of interest, where \({\tilde{{\mathcal{R}}}}_{i,j}\) denotes the \({j}{\rm{th}}\) tensor factor of the \({i}{\rm{th}}\) gate. Creating an effective Pauli channel per application enables errors to be systematically amplified under \(m\)-fold iterations for more precise and SPAM-free estimation of the errors in the interleaved red gates \(\tilde{{\mathcal{G}}}\). The blue and the red gates together form the random circuit \(\tilde{{\mathcal{C}}}\). The sequence of local operations before the first and last rounds of random Pauli cycles are identified as conceptually distinct but were compiled into the initial and final round of local gates in the experiment. The experimental parameters \(K,m\), and \(L\) of this work and the exact definitions of \(\tilde{{\mathcal{B}}}\) and \(\tilde{{\mathcal{R}}}\) are given in Supplementary Note 7.